In the next section, the Bark frequency scale is reviewed, followed by a section reviewing the bilinear transformation and its specialization to the first-order allpass transformation. Section IV is concerned with optimally choosing the allpass parameter: A weighted equation-error solution is derived which is shown to be essentially equal to the optimal least-squares solution. The optimal Chebyshev solution is compared and found to be insignificantly different from the least-squares solutions. A variation on the error criterion which is only concerned with mapped bandwidth error, as opposed to the absolute error in the mapping of Hz to Barks, is introduced and evaluated. A simple closed-form expression relating the sampling rate to the optimal warping parameter is presented. Section IV concludes with a filter-design example illustrating the benefits of working over a Bark frequency scale. Finally, Section V applies the methods of Section IV to approximating the ERB scale, and the results, while potentially useful, are found to be significantly less accurate than for the Bark scale. The paper concludes with a summary of findings.